A325271 Number of integer partitions of n with frequency depth round(sqrt(n)).

1, 1, 1, 1, 2, 1, 3, 4, 6, 8, 11, 11, 19, 44, 53, 63, 83, 113, 124, 171, 190, 344, 429, 525, 662, 796, 981, 1182, 1442, 1709, 2096, 2663, 3406, 4315, 5426, 6784, 8417, 10466, 12824, 15721, 19104, 23267, 27981, 5, 14, 36, 76, 143, 269, 446, 738, 1143, 1754

Offset: 0

Author

Gus Wiseman, Apr 23 2019

Keywords

• nonn

Comments

The adjusted frequency depth of an integer partition is 0 if the partition is empty, and otherwise it is 1 plus the number of times one must take the multiset of multiplicities to reach a singleton. For example, the partition (32211) has adjusted frequency depth 5 because we have: (32211) -> (221) -> (21) -> (11) -> (2).

Examples

			The a(2) = 1 through a(10) = 11 partitions:
  (2)  (111)  (22)    (11111)  (33)      (43)   (53)    (54)      (64)
              (1111)           (222)     (52)   (62)    (63)      (73)
                               (111111)  (61)   (71)    (72)      (82)
                                         (421)  (431)   (81)      (91)
                                                (521)   (432)     (532)
                                                (3311)  (531)     (541)
                                                        (621)     (631)
                                                        (222111)  (721)
                                                                  (3322)
                                                                  (4321)
                                                                  (4411)
		

Crossrefs

Cf. A117571, A181819, A225485, A323014, A323023, A325245, A325246, A325272, A325253, A325258, A325278, A325280, A325282.

Programs

Mathematica

fdadj[ptn_List]:=If[ptn=={},0,Length[NestWhileList[Sort[Length/@Split[#]]&,ptn,Length[#]>1&]]];
Table[Length[Select[IntegerPartitions[n],fdadj[#]==Round[Sqrt[n]]&]],{n,0,30}]